SPH3U UNIVERSITY PHYSICS
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1 SPH3U UNIVERSITY PHYSICS REVIEW: MATH SKILLS L (P.651; 653) Many people believe that all measurements are reliable (consistent over many trials), precise (to as many decimal places as possible), and accurate (representing the actual value). But there are many things that can go wrong when measuring. For example: 1 There may be limitations that make the instrument or its use unreliable (inconsistent). The investigator may make a mistake or fail to follow the correct techniques when reading the measurement to the available precision (number of decimal places). The instrument may be faulty or inaccurate; a similar instrument may give different readings. 2 1
2 1. What three things can you do during an experiment to help eliminate errors? 1. To be sure that you have measured correctly, you should repeat your measurements at least three times. 2. If your measurements appear to be reliable, calculate the mean and use that value. 3. To be more precise about the accuracy, repeat the measurements with a different instrument There are two types of measurement error. What are they? random error and systematic error 4 Random Error Random error results when an estimate is made to obtain the last digit for any measurement. The size of the random error is determined by the precision of the measuring instrument. For example, when measuring length with a measuring tape, it is necessary to estimate between the marks on the measuring tape. If these marks are 1 cm apart, the random error will be greater and the precision will be less than if the marks are 1 mm apart. Such errors can be reduced by taking the average of several readings. RANDOM ERROR results when the last digit is estimated reduced by taking the average of several readings 5 2
3 Systematic Error Systematic error is associated with an inherent problem with the measuring system, such as the presence of an interfering substance, incorrect calibration, or room conditions. For example, if a balance is not zeroed at the beginning, all measurements will have a systematic error; using a slightly worn metre stick will also introduce error. Such errors are reduced by adding or subtracting the known error or calibrating the instrument. SYSTEMATIC ERROR due to a problem with the measuring device reduced by adding/subtracting the error or calibrating the device 6 Accuracy & Precision In everyday usage, "accuracy" and "precision" are used interchangeably to describe how close a measurement is to a true value, but in science it is important to make a distinction between them. Accuracy refers to how close a value is to its accepted value. Precision is the place value of the last measureable digit. ACCURACY how close a value is to its accepted value PRECISION place value of last measureable digit 7 Accuracy & Precision For example, the position of the darts in each of the figures are analogous to measured or calculated results in a laboratory setting. The results in (a) are precise and accurate, in (b) they are precise but not accurate, and in (c) they are neither precise nor accurate. 8 3
4 Percentage Error No matter how precise a measurement is, it still may not be accurate. The percentage error is the absolute value of the difference between experimental and accepted values expressed as a percentage of the accepted value. % error = experimental value accepted value accepted value x 100 NOTE! The bars ( ) in the equation above represent absolute value. means that, mathematically, if a = 3 and b = -3 then a = b = 3. This 9 Percentage Difference Sometimes if two values of the same quantity are measured, it is useful to compare the precision of these values by calculating the percentage difference between them. measurement 1 measurement 2 % difference = x 100 measurement 1 + measurement 2 2 NOTE! Magnitude is a term frequently used by physicists. The magnitude of a quantity is the same as its absolute value At a certain location the acceleration due to gravity is 9.82 m/s 2 [down]. Calculate the percentage error of the following experimental values of g at that location. (a) 8.94 m/s 2 [down] (b) 9.95 m/s 2 [down] (a) 8.96 (b)
5 4. Calculate the percentage difference between the two experimental values (8.94 m/s 2 and 9.95 m/s 2 ) used in question # U Check Your Learning WIKI (REVIEW) 3UR - WS#1 (Math Skills) 3UR - QUIZ#1 (Math Skills - Part 1) 13 5
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